Studies on the analytic behaviour of number theoretic L-functions

数论L-函数解析行为的研究

• 批准号: 12440004
• 负责人: MATSUMOTO Kohji
• 金额: $ 8.77万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440004/
• 关键词: Multiple zeta-function; Euler-Zagier sum; Analytic continuation; Hurwitx zeta-function; Ramanujan's formula; Modular relation; Universality; Automorphic L-function; 漸近展開; Ramanujan formula; 近似関数等式; Mellin-Barnes formula; universality; Rankin-

(1) We introduced the notion of generalized multiple zeta-functions, which is a generalization of both the Euler-Zagier multiple sums and the Barnes multiple zeta-functions, and, by using the Mellin-Barnes integral formula, proved their analytic continuation and asymptotic expansions. As applications, we proved asymptotic expansions of higher power moments of Hurwitz zeta-functions, and also explicit formulas of determinants of the Laplacians of high-dimensional spheres.(2) We found a basic principle connecting Ramanujan's formula, modular relations, and approximate functional equations, and proved rapidly converget series expressions of various L-functions, in connection with multiple zeta-functions.(3) We introduced the positive density method in universality theory, and proved the universality of automorphic L-functions, and Rankin-Selberg L-fanctions, attached to cusp forms of SL (2, II) or its congruence sabgroups, also the joint universality of Lerch zeta-functions.

(1)引入了广义重zeta函数的概念，它是Euler-Zagier重和与巴恩斯重zeta函数的推广，并利用Mellin-巴恩斯积分公式证明了它们的解析延拓与渐近展开.作为应用，我们证明了Hurwitz zeta函数的高次幂矩的渐近展开式，以及高维球面的Laplacian行列式的显式公式。(2)找到了Ramanujan公式、模关系和近似函数方程之间的基本原理，证明了在多个zeta函数的情况下，各种L函数的级数表达式可以快速收敛。(3)引入普适性理论中的正密度方法，证明了SL（2，II）的尖点形式或其同余子群上的自守L-函数、Rankin-Selberg L-函数的普适性，以及Lerch zeta-函数的联合普适性.

项目成果

K.Sato: "Finiteness of a certain motivic cohomology group of varieties over local and global fields"Class Field Theory. 325-333 (2001)

K.Sato：“局部和全局域上簇的某个动机上同调群的有限性”类域论。

S.Kanemitsu, H.Kumagai, M.Yoshimoto: "On rapidly convergent series expressions for zeta- and L-values, and log sine integrals"ibid.. 91-104

S.Kanemitsu、H.Kumagai、M.Yoshimoto：“关于 zeta 值和 L 值以及对数正弦积分的快速收敛级数表达式”同上.. 91-104

S.Akiyama: "On the boundary of self affine filings generated by Pisot numbers"J.Math.Soc.Japan. 54. 283-308 (2002)

S.Akiyama：“关于皮索数生成的自仿射归档的边界”J.Math.Soc.Japan。

K.Matsumoto, L.Weng: "Zeta-functions defined by two polynomials, Number Theoretic Methods-Future Trends"Kluwer. 233-262 (2002)

K.Matsumoto、L.Weng：“由两个多项式定义的 Zeta 函数，数论方法 - 未来趋势”Kluwer。

S.Kanemitsu, Y.Tanigawa, M.Yoshimoto: "On rapidly convergent series for Dirichlet L-function values via the modular relation, Number Theory and Discrete Math."Hindustan/Birkhauser. 113-133 (2002)

S.Kanemitsu、Y.Tanikawa、M.Yoshimoto：“通过模关系、数论和离散数学实现狄利克雷 L 函数值的快速收敛级数。”Hindustan/Birkhauser。